Twisted Alexander norms give lower bounds on the Thurston norm

TL;DR

This paper introduces twisted Alexander norms for 3-manifolds, demonstrating they provide lower bounds on the Thurston norm and enabling the complete determination of the Thurston norm in cases where previous norms were insufficient.

Contribution

The paper generalizes existing norms to twisted Alexander norms and shows they can determine the Thurston norm for many 3-manifolds beyond prior methods.

Findings

Twisted Alexander norms give effective lower bounds on the Thurston norm.

These norms can fully determine the Thurston norm in cases where McMullen and Turaev norms cannot.

The approach extends the toolkit for analyzing 3-manifold topology.

Abstract

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number bigger than one generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which can not be determined by norms of McMullen and Turaev.